How Good is Almost Perfect? Malte Helmert Gabriele R oger - PowerPoint PPT Presentation

How Good is Almost Perfect? Malte Helmert Gabriele R¨ oger Albert-Ludwigs-Universit¨ at Freiburg, Germany AAAI 2008

Introduction Theoretical Results Experimental Results Conclusion Outline Introduction 1 Theoretical Results 2 Experimental Results 3 Conclusion 4

Introduction Theoretical Results Experimental Results Conclusion Optimal sequential planning Optimal sequential planning = A ∗ (or similar) + admissible heuristic (mostly)

Introduction Theoretical Results Experimental Results Conclusion Folklore Everybody knows: If a heuristic has constant absolute error, A ∗ requires a linear number of node expansions.

Introduction Theoretical Results Experimental Results Conclusion Comparison: Heuristic vs. breadth-first search Actually, state-of-the art optimal sequential planners are not much better than breadth-first search. Experiments of Helmert, Haslum & Hoffmann (2007) BFHSP solved 37 tasks A ∗ + h max solved 46 tasks A ∗ + h PDB solved 54 tasks blind search solved 42 tasks

Introduction Theoretical Results Experimental Results Conclusion Are our heuristics bad? Two possible explanations: Our heuristics aren’t that good. There is something fishy going on. (Or both.)

Introduction Theoretical Results Experimental Results Conclusion Folklore + fine print Everybody knows: If a heuristic has constant absolute error, A ∗ requires a linear number of node expansions. But. . . This relies on several assumptions: fixed branching factor only a single goal state no transpositions These assumptions do not hold in any common planning task!

Introduction Theoretical Results Experimental Results Conclusion Almost perfect heuristics Almost perfect heuristics differ from the perfect heuristic h ∗ only by an additive constant: Definition Define heuristic h ∗ − c (for c ∈ N 1 ) as ( h ∗ − c )( s ) := max( h ∗ ( s ) − c, 0) → unlikely to be obtainable in practice

Introduction Theoretical Results Experimental Results Conclusion The topic of this work How many nodes must A ∗ expand for a planning task T , given an almost perfect heuristic h ∗ − c ? Definition N c ( T ) := number of states s with g ( s ) + ( h ∗ − c )( s ) < h ∗ ( T ) � If this number grows fast with scaling task size, � we have a problem. Objective Results for N c ( T ) for IPC domains → Focus on domains in APX

Introduction Theoretical Results Experimental Results Conclusion Outline Introduction 1 Theoretical Results 2 Experimental Results 3 Conclusion 4

Introduction Theoretical Results Experimental Results Conclusion Our goal Find sequence ( T n ) of scaling tasks for which N c ( T n ) grows exponentially, even for small values of c .

Introduction Theoretical Results Experimental Results Conclusion Gripper initial state goal state T n : Task with n balls S n : Total number of reachable states of T n S n = 2 · (2 n + 2 n 2 n − 1 + n ( n − 1)2 n − 2 )

Introduction Theoretical Results Experimental Results Conclusion Gripper Theorem Theorem Let n ∈ N 0 with n ≥ 3 . If n is even, then N 1 ( T n ) = N 2 ( T n ) = 1 2 S n − 3 N c ( T n ) = S n − 2 n − 2 for all c ≥ 3 . If n is odd, then N 1 ( T n ) = N 2 ( T n ) = S n − 3 N c ( T n ) = S n − 2 for all c ≥ 3 .

Introduction Theoretical Results Experimental Results Conclusion Gripper Proof Proof sketch n is even states with an even number of balls in each room basically all are part of an optimal plan states with an odd number of balls in each room all are part of plans of length h ∗ ( T n ) + 2 n is odd basically all states are part of an optimal plan

Introduction Theoretical Results Experimental Results Conclusion Miconic-Simple-Adl initial state goal state T n : Task with n passengers (and n + 1 floors) S n : Total number of reachable states of T n S n = 3 n ( n + 1)

Introduction Theoretical Results Experimental Results Conclusion Miconic-Simple-Adl Theorem Theorem For all c ≥ 4 : N c ( T n ) = S n − (2 n − 1)( n + 1) .

Introduction Theoretical Results Experimental Results Conclusion Blocksworld initial state goal state T n : Task with n blocks ( n ≥ 2 )

Introduction Theoretical Results Experimental Results Conclusion Blocksworld Theorem Theorem n − 3 � N 1 ( T n ) = 4 · B k + 3 B n − 2 + 1 k =0 N 1 ( T n ) N 1 ( T n ) n n 2 4 9 3748 3 8 10 17045 4 15 11 84626 5 32 12 453698 6 82 13 2605383 7 253 14 15924744 8 914 15 103071652

Introduction Theoretical Results Experimental Results Conclusion Outline Introduction 1 Theoretical Results 2 Experimental Results 3 Conclusion 4

Introduction Theoretical Results Experimental Results Conclusion Question Theoretical results There exist task families for which the number of states expanded by h ∗ − c grows exponentially, even for small c . Interesting question Can we observe this behaviour in practice? → Experiments with IPC tasks

Introduction Theoretical Results Experimental Results Conclusion Problem Problem Values N c ( T ) are defined in terms of h ∗ . Usually h ∗ cannot be determined efficiently. Naive way of computing N c ( T ) Completely explore the state space of T . Search backwards from the goals to determine the h ∗ ( s ) values. → Observation: Generating all states is not necessary.

Introduction Theoretical Results Experimental Results Conclusion Search space goal node node belongs to N c ( T ) node does not belong to N c ( T ) h ∗ ( T ) h ∗ ( T ) + c − 1 � Poster session: today, 6:00-9:30 PM

Introduction Theoretical Results Experimental Results Conclusion Results Blocksworld N 1 ( T ) N 2 ( T ) N 3 ( T ) N 4 ( T ) N 5 ( T ) task h ∗ ( T ) 04-1 10 10 10 16 16 29 05-2 16 28 28 72 72 162 06-2 20 27 27 144 144 476 07-1 22 106 106 606 606 2244 08-1 20 66 66 503 503 2440 09-0 30 411 411 3961 3961 21135

Introduction Theoretical Results Experimental Results Conclusion Results Gripper N 1 ( T ) N 2 ( T ) N 3 ( T ) N 4 ( T ) task h ∗ ( T ) 01 11 125 125 246 246 02 17 925 925 1842 1842 03 23 5885 5885 11758 11758 04 29 34301 34301 68586 68586 05 35 188413 188413 376806 376806 06 41 991229 991229 1982434 1982434 07 47 5046269 5046269 10092510 10092510

Introduction Theoretical Results Experimental Results Conclusion Results Logistics (IPC 2) task h ∗ ( T ) N 1 ( T ) N 2 ( T ) N 3 ( T ) N 4 ( T ) N 5 ( T ) 4-0 20 159 408 1126 1780 2936 5-0 27 459 2391 5693 14370 21124 6-0 25 411 2160 5712 14485 23967 7-1 44 17617 111756 427944 1173096 8-1 44 4843 27396 157645 558869 9-0 36 2778 15878 61507 183826 460737 10-0 45 10847 11-0 48 10495

Introduction Theoretical Results Experimental Results Conclusion Results Miconic-Simple-Adl N 1 ( T ) N 2 ( T ) N 3 ( T ) N 4 ( T ) N 5 ( T ) task h ∗ ( T ) 1-0 4 4 4 4 4 4 2-1 6 6 22 26 26 26 3-1 10 58 102 102 102 102 4-2 14 148 280 470 560 560 5-1 15 209 759 1136 1326 1399 6-4 18 397 948 1936 2844 3436 7-4 23 3236 7654 11961 15780 16968 8-3 24 1292 5870 15188 25914 34315 9-3 28 20891 39348 39348 39348 39348 10-3 28 6476 16180 65477 129400 224495 11-3 32 58268 130658 258977 399850 497030 12-4 34 83694 181416 541517 970632 1640974 13-2 40 461691 947674 2203931 3443154 4546823

Introduction Theoretical Results Experimental Results Conclusion Results Miconic-Strips N 1 ( T ) N 2 ( T ) N 3 ( T ) N 4 ( T ) N 5 ( T ) task h ∗ ( T ) 1-0 4 4 4 4 4 4 2-1 7 18 29 34 37 37 3-1 11 70 138 195 241 251 4-4 15 166 507 814 1182 1348 5-4 18 341 1305 2708 4472 5933 6-4 21 509 2690 7086 13657 21177 7-4 25 3668 13918 32836 61852 95548 8-3 28 4532 35529 97529 205009 349491 9-3 32 25265 114840 321202 700640 1239599 10-3 34 8150 97043 423641 1151402 2505892

Introduction Theoretical Results Experimental Results Conclusion Introduction 1 Theoretical Results 2 Experimental Results 3 Conclusion 4

Introduction Theoretical Results Experimental Results Conclusion Dismal prospects Depressing theoretical and experimental results Other (similar) search techniques cannot perform better than A ∗ . With other (real) heuristics it gets worse.

Introduction Theoretical Results Experimental Results Conclusion What is the cause of this behaviour? Main problem many independently solvable subproblems which can be arbitrarily permuted many possible orders Why is this not common knowledge? → does not happen in 15-Puzzle, Rubik’s Cube, etc.

Introduction Theoretical Results Experimental Results Conclusion What do the results mean for us? Some possible conclusions: Conclusion? We need heuristics that are better than almost perfect. How feasible is this? Conclusion? We need more search enhancements. Look to domain-dependent search for guidance?